A Fock space model for addition and multiplication of c-free random variables

Popa, Mihai

doi:10.1090/S0002-9939-2014-11786-2

A Fock space model for addition and multiplication of c-free random variables

Author: Mihai Popa
Journal: Proc. Amer. Math. Soc. 142 (2014), 2001-2012
MSC (2010): Primary 46L54, 30H20
DOI: https://doi.org/10.1090/S0002-9939-2014-11786-2
Published electronically: February 27, 2014
MathSciNet review: 3182019
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The paper presents a Fock space model suitable for construction of c-free algebras. Immediate applications are direct proofs for the properties of the c-free - and -transforms.

[1] Serban Teodor Belinschi, C-free convolution for measures with unbounded support, Von Neumann algebras in Sibiu, Theta Ser. Adv. Math., vol. 10, Theta, Bucharest, 2008, pp. 1–7. MR 2512322
[2] Anis Ben Ghorbal and Michael Schürmann, Quantum stochastic calculus on Boolean Fock space, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 7 (2004), no. 4, 631–650. MR 2105916, https://doi.org/10.1142/S0219025704001815
[3] Florin Boca, Free products of completely positive maps and spectral sets, J. Funct. Anal. 97 (1991), no. 2, 251–263. MR 1111181, https://doi.org/10.1016/0022-1236(91)90001-L
[4] Marek Bożejko, Uniformly bounded representations of free groups, J. Reine Angew. Math. 377 (1987), 170–186. MR 887407, https://doi.org/10.1515/crll.1987.377.170
[5] Marek Bożejko, Positive definite functions on the free group and the noncommutative Riesz product, Boll. Un. Mat. Ital. A (6) 5 (1986), no. 1, 13–21 (English, with Italian summary). MR 833375
[6] Marek Bożejko and Roland Speicher, 𝜓-independent and symmetrized white noises, Quantum probability & related topics, QP-PQ, VI, World Sci. Publ., River Edge, NJ, 1991, pp. 219–236. MR 1149828
[7] Marek Bożejko, Michael Leinert, and Roland Speicher, Convolution and limit theorems for conditionally free random variables, Pacific J. Math. 175 (1996), no. 2, 357–388. MR 1432836
[8] Kenneth J. Dykema, Multilinear function series and transforms in free probability theory, Adv. Math. 208 (2007), no. 1, 351–407. MR 2304321, https://doi.org/10.1016/j.aim.2006.02.011
[9] Etienne F. Blanchard and Kenneth J. Dykema, Embeddings of reduced free products of operator algebras, Pacific J. Math. 199 (2001), no. 1, 1–19. MR 1847144, https://doi.org/10.2140/pjm.2001.199.1
[10] Uffe Haagerup, On Voiculescu’s 𝑅- and 𝑆-transforms for free non-commuting random variables, Free probability theory (Waterloo, ON, 1995) Fields Inst. Commun., vol. 12, Amer. Math. Soc., Providence, RI, 1997, pp. 127–148. MR 1426838
[11] Anna Dorota Krystek, Infinite divisibility for the conditionally free convolution, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 10 (2007), no. 4, 499–522. MR 2376439, https://doi.org/10.1142/S0219025707002919
[12] Wojciech Młotkowski, Operator-valued version of conditionally free product, Studia Math. 153 (2002), no. 1, 13–30. MR 1948925, https://doi.org/10.4064/sm153-1-2
[13] Alexandru Nica and Roland Speicher, Lectures on the combinatorics of free probability, London Mathematical Society Lecture Note Series, vol. 335, Cambridge University Press, Cambridge, 2006. MR 2266879
[14] Mihai Popa, Multilinear function series in conditionally free probability with amalgamation, Commun. Stoch. Anal. 2 (2008), no. 2, 307–322. MR 2446696, https://doi.org/10.31390/cosa.2.2.09
[15] Mihai Popa, Realization of conditionally monotone independence and monotone products of completely positive maps, J. Operator Theory 68 (2012), no. 1, 257–274. MR 2966045
[16] Mihai Popa, Non-crossing linked partitions and multiplication of free random variables, Operator theory live, Theta Ser. Adv. Math., vol. 12, Theta, Bucharest, 2010, pp. 135–143. MR 2731869
[17] Mihai Popa and Victor Vinnikov, Non-commutative functions and the non-commutative free Lévy-Hinčin formula, Adv. Math. 236 (2013), 131–157. MR 3019719, https://doi.org/10.1016/j.aim.2012.12.013
[18] Mihai Popa and Jiun-Chau Wang, On multiplicative conditionally free convolution, Trans. Amer. Math. Soc. 363 (2011), no. 12, 6309–6335. MR 2833556, https://doi.org/10.1090/S0002-9947-2011-05242-6
[19] Dan Voiculescu, Addition of certain noncommuting random variables, J. Funct. Anal. 66 (1986), no. 3, 323–346. MR 839105, https://doi.org/10.1016/0022-1236(86)90062-5
[20] Dan Voiculescu, Multiplication of certain noncommuting random variables, J. Operator Theory 18 (1987), no. 2, 223–235. MR 915507
[21] D. V. Voiculescu, K. J. Dykema, and A. Nica, Free random variables, CRM Monograph Series, vol. 1, American Mathematical Society, Providence, RI, 1992. A noncommutative probability approach to free products with applications to random matrices, operator algebras and harmonic analysis on free groups. MR 1217253
[22] Jiun-Chau Wang, Limit theorems for additive conditionally free convolution, Canad. J. Math. 63 (2011), no. 1, 222–240. MR 2779139, https://doi.org/10.4153/CJM-2010-075-4